Golden Rule savings rate
Encyclopedia
In economics
, the Golden Rule savings rate is the rate of savings which maximizes steady state level or growth of consumption , as for example in the Solow growth model. Although the concept can be found earlier in John von Neumann
and Maurice Allais
's works, the term is generally attributed to Edmund Phelps
who wrote in 1961 that the Golden Rule "do unto others as you would have them do unto you" could be applied inter-generationally inside the model to arrive at some form of "optimum
".
In the Solow growth model, a steady state savings rate of 100% implies that all income is going to investment capital
for future production, implying a steady state consumption level of zero. A savings rate of 0% implies that no new investment capital is being created, so that the capital stock depreciates without replacement. This makes a steady state unsustainable except at zero output, which again implies a consumption level of zero. Somewhere in between is the "Golden Rule" level of savings, where the savings propensity is such that per-capita consumption
is at its maximum possible constant value.
To discover the optimal capital/labour ratio, and thus the golden rule savings rate, first note that consumption can be seen as the residual output that remains after providing for the investment that maintains steady state:
Differential calculus
methods can identify which steady state value for the capital/labour ratio maximises per capita consumption. The golden rule savings rate is then implied by the connection between s and k in steady state (see above).
In detail, if
is the golden rule steady state level of k, then 
requires 
, i.e. 
The Inada conditions ensure that this rule is satisfied by a unique
and thus produces a unique 
. Since steady state requires a particular level of investment, i.e. saved output: 
, then the golden rule savings rate must be whatever is required to generate this;
Given the rule for optimal k, this may also be expressed as
in which
is the marginal product of capital ( 
) at the optimal value of k and 
is the corresponding average product of capital ( 
)
The actual values of
, 
, 
, and 
depend upon the precise specification of the production function
. For example, a Cobb-Douglas
specification with constant returns to scale has
, hence 
and 
. This gives 
and hence 
, 
.
es, for example, may reduce the level of consumption and increase the savings rate, whereas capital gains tax
es may reduce the savings rate. These policies are often known as savings incentives in the west
, where it is felt that the prevailing savings rate is "too low" (below the Golden Rule rate), and consumption incentives in countries like Japan
where demand
is widely considered to be too weak because the savings rate is "too high" (above the Golden Rule).
(if the revenue from bond
sales is spent on present government consumption rather than infrastructure
development, say).
to be permanent then it is hard to reconcile the common hypothesis that rising rates discourage consumption with rational expectations
(since the ultimate purpose of saving is consumption). However, consumption taxes tend to vary (e.g. with changes in government or movement between countries), and so currently high consumption taxes may be expected to go away at some point in the future, creating an increased incentive for saving. The efficient level of capital income tax in the steady state has been studied in the context of a general equilibrium
model and Judd (1985) has shown that the optimal tax rate is zero.. However, Chamley (1986) says that in reaching the steady state (in the short run) a high capital income tax is an efficient revenue source.
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, the Golden Rule savings rate is the rate of savings which maximizes steady state level or growth of consumption , as for example in the Solow growth model. Although the concept can be found earlier in John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
and Maurice Allais
Maurice Allais
Maurice Félix Charles Allais was a French economist, and was the 1988 winner of the Nobel Memorial Prize in Economics "for his pioneering contributions to the theory of markets and efficient utilization of resources."...
's works, the term is generally attributed to Edmund Phelps
Edmund Phelps
Edmund Strother Phelps, Jr. is an American economist and the winner of the 2006 Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel. Early in his career he became renowned for his research at Yale's Cowles Foundation in the first half of the 1960s on the sources of economic growth...
who wrote in 1961 that the Golden Rule "do unto others as you would have them do unto you" could be applied inter-generationally inside the model to arrive at some form of "optimum
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
".
In the Solow growth model, a steady state savings rate of 100% implies that all income is going to investment capital
Capital (economics)
In economics, capital, capital goods, or real capital refers to already-produced durable goods used in production of goods or services. The capital goods are not significantly consumed, though they may depreciate in the production process...
for future production, implying a steady state consumption level of zero. A savings rate of 0% implies that no new investment capital is being created, so that the capital stock depreciates without replacement. This makes a steady state unsustainable except at zero output, which again implies a consumption level of zero. Somewhere in between is the "Golden Rule" level of savings, where the savings propensity is such that per-capita consumption
Consumption (economics)
Consumption is a common concept in economics, and gives rise to derived concepts such as consumer debt. Generally, consumption is defined in part by comparison to production. But the precise definition can vary because different schools of economists define production quite differently...
is at its maximum possible constant value.
Derivation of the Golden Rule savings rate
The following arguments are presented more completely in Chapter 1 of Barro and Sala-i-Martin , in texts such as Abel et al. , and also in the Growth Theory website at newschool.edu
Let k be the capital/labour ratio (i.e. capitalCapital (economics)In economics, capital, capital goods, or real capital refers to already-produced durable goods used in production of goods or services. The capital goods are not significantly consumed, though they may depreciate in the production process...
per capita), y be the resulting per capita output ( 
), and s be the savings rate. The steady state is defined as a situation in which per capita output is unchanging, which implies that k be constant. This requires that the amount of saved output be exactly what is needed to (1) equip any additional workers and (2) replace any worn out capital.
In a steady state, therefore: 
, where n is the constant exogenous population growth rate, and d is the constant exogenous rate of depreciation of capital. Since n and d are constant and 
satisfies the Inada conditionsInada conditionsIn macroeconomics, the Inada conditions are assumptions about the shape of a production function that guarantee the stability of an economic growth path in a neoclassical growth model....
, this expression may be read as an equation connecting s and k in steady state: any choice of s implies a unique value for k (thus also for y) in steady state. Since consumption is proportional to output ( 
), then a choice of value for s implies a unique level of steady state per capita consumption. Out of all possible choices for s, one will produce the highest possible steady state value for c and is called the golden rule savings rate.
An important question for policy-makers is whether the economy is saving too much or too little. Given the interconnection of s and k in steady state, noted above, the question can be phrased: "How much capital per worker (k) is needed to achieve the maximum level of consumption per worker in the steady state?"
To discover the optimal capital/labour ratio, and thus the golden rule savings rate, first note that consumption can be seen as the residual output that remains after providing for the investment that maintains steady state:
Differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
methods can identify which steady state value for the capital/labour ratio maximises per capita consumption. The golden rule savings rate is then implied by the connection between s and k in steady state (see above).
In detail, if
is the golden rule steady state level of k, then requires , i.e. The Inada conditions ensure that this rule is satisfied by a unique
and thus produces a unique . Since steady state requires a particular level of investment, i.e. saved output: , then the golden rule savings rate must be whatever is required to generate this;Given the rule for optimal k, this may also be expressed as
in which
is the marginal product of capital ( ) at the optimal value of k and is the corresponding average product of capital ( )The actual values of
, , , and depend upon the precise specification of the production functionProduction function
In microeconomics and macroeconomics, a production function is a function that specifies the output of a firm, an industry, or an entire economy for all combinations of inputs...
. For example, a Cobb-DouglasCobb-Douglas
In economics, the Cobb–Douglas functional form of production functions is widely used to represent the relationship of an output to inputs. Similar functions were originally used by Knut Wicksell , while the Cobb-Douglas form was developed and tested against statistical evidence by Charles Cobb and...
specification with constant returns to scale has
, hence and . This gives and hence , .Policy that can change the savings rate
Various economic policies can have an effect on the savings rate and, given data about whether an economy is saving too much or too little, can in turn be used to approach the Golden Rule level of savings. Consumption taxConsumption tax
A consumption tax is a tax on spending on goods and services. The tax base of such a tax is the money spent on consumption. Consumption taxes are usually indirect, such as a sales tax or a value added tax...
es, for example, may reduce the level of consumption and increase the savings rate, whereas capital gains tax
Capital gains tax
A capital gains tax is a tax charged on capital gains, the profit realized on the sale of a non-inventory asset that was purchased at a lower price. The most common capital gains are realized from the sale of stocks, bonds, precious metals and property...
es may reduce the savings rate. These policies are often known as savings incentives in the west
Western world
The Western world, also known as the West and the Occident , is a term referring to the countries of Western Europe , the countries of the Americas, as well all countries of Northern and Central Europe, Australia and New Zealand...
, where it is felt that the prevailing savings rate is "too low" (below the Golden Rule rate), and consumption incentives in countries like Japan
Japan
Japan is an island nation in East Asia. Located in the Pacific Ocean, it lies to the east of the Sea of Japan, China, North Korea, South Korea and Russia, stretching from the Sea of Okhotsk in the north to the East China Sea and Taiwan in the south...
where demand
Demand
- Economics :*Demand , the desire to own something and the ability to pay for it*Demand curve, a graphic representation of a demand schedule*Demand deposit, the money in checking accounts...
is widely considered to be too weak because the savings rate is "too high" (above the Golden Rule).
Private and public saving
Japan's high rate of private saving is offset by its high public debt. A simple approximation of this is that the government has borrowed 100% of GDP from its own citizens backed only with the promise to pay from future taxation. This does not necessarily lead to capital formation through investmentInvestment
Investment has different meanings in finance and economics. Finance investment is putting money into something with the expectation of gain, that upon thorough analysis, has a high degree of security for the principal amount, as well as security of return, within an expected period of time...
(if the revenue from bond
Bond (finance)
In finance, a bond is a debt security, in which the authorized issuer owes the holders a debt and, depending on the terms of the bond, is obliged to pay interest to use and/or to repay the principal at a later date, termed maturity...
sales is spent on present government consumption rather than infrastructure
Infrastructure
Infrastructure is basic physical and organizational structures needed for the operation of a society or enterprise, or the services and facilities necessary for an economy to function...
development, say).
Golden rule taxes within economic models
If consumption tax rates are expectedExpectation
In the case of uncertainty, expectation is what is considered the most likely to happen. An expectation, which is a belief that is centered on the future, may or may not be realistic. A less advantageous result gives rise to the emotion of disappointment. If something happens that is not at all...
to be permanent then it is hard to reconcile the common hypothesis that rising rates discourage consumption with rational expectations
Rational expectations
Rational expectations is a hypothesis in economics which states that agents' predictions of the future value of economically relevant variables are not systematically wrong in that all errors are random. An alternative formulation is that rational expectations are model-consistent expectations, in...
(since the ultimate purpose of saving is consumption). However, consumption taxes tend to vary (e.g. with changes in government or movement between countries), and so currently high consumption taxes may be expected to go away at some point in the future, creating an increased incentive for saving. The efficient level of capital income tax in the steady state has been studied in the context of a general equilibrium
General equilibrium
General equilibrium theory is a branch of theoretical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium, hence general...
model and Judd (1985) has shown that the optimal tax rate is zero.. However, Chamley (1986) says that in reaching the steady state (in the short run) a high capital income tax is an efficient revenue source.